Variational Representations for Continuous Time Processes

A variational formula for positive functionals of a Poisson random measure and Brownian motion is proved. The formula is based on the relative entropy representation for exponential integrals, and can be used to prove large deviation type estimates. A general large deviation result is proved, and illustrated with an example. 1 Introduction In this paper we prove a variational representation for positive measurable functionals of a Poisson random measure and an in…nite dimensional Brownian motion. These processes provide the driving noises for a wide range of important process models in continuous time, and thus we also obtain variational representations for these processes when a strong solution exists. The representations have a number of uses, the most important being to prove large deviation estimates. The theory of large deviations is by now well understood in many settings, but there remain some situations where the topic is not as well developed. These are often settings where technical issues challenge standard approaches, and the problem of …nding nearly optimal or even reasonably weak su¢ cient conditions is hindered as much by technique of proof as any other issue. Variational representations of the sort developed in this paper have been shown to be particularly useful, when combined with weak convergence methods, for analyzing such systems. For example, Brownian motion representations have been used by [1, 6, 7, 8, 10, 11, 18, 19, 21, 22, 24, 25, 26, 29, 30] in the large deviation analysis of solutions to SPDEs in the small noise limit, and recently in [5] for interacting particle limits. Other Department of Statistics and Operations Research, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA. Research supported in part by the Army Research O¢ ce (Grant W911NF-0-1-0080). yLefschetz Center for Dynamical Systems, Division of Applied Mathematics, Brown University, Providence, RI, 02912, USA. Research supported in part by the National Science Foundation, the Army Research O¢ ce, and the Air Force O¢ ce of Scienti…c Research. zInstitute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN 55455, USA